Squat effects on high speed craft in restricted waterways

Ocean Engineering 33 (2006) 365–381 www.elsevier.com/locate/oceaneng

Squat effects on high speed craft in restricted waterways K.S. Varyani* Department of Naval Architecture and Marine Engineering, Universities of Glasgow and Strathclyde, Henry Dyer Building, 100 Montrose Street, Glasgow G4 OLZ, UK Received 12 January 2005; accepted 20 April 2005 Available online 8 August 2005

Abstract The vessels considered here for the squat studies are a bulk carrier and an High Speed Craft (HSC). The bulk carrier of full form (CBZ0.81) is used for validation purpose and subsequently the numerical computations are performed for a High Speed Craft with fine hull form of CBZ0.467. A high speed ferry, with LCB and LCF quite aft of midship is considered for squat study. For a vessel speed of above 6.0 knots and for a waterway width greater than the vessel length there appears to be a sudden increase in sinkage at the stern accompanied by a large value of bow emergence. It could be seen that the speed of the vessel has a much greater influence on the sinkage and trim of the vessel than the waterway restrictions. q 2005 Elsevier Ltd. All rights reserved. Keywords: Squat; High speed craft; Restricted waterways

1. Introduction The phenomenon of increased immersion and trim of ships when underway in water compared to still water floating condition, also referred as ‘Squat’ is known to mariners for many centuries. However, the magnitude of this additional sinkage with respect to the vessel speed and waterway restrictions was not clearly understood for a long time. The need to quantify the same has gained importance with the development of several new specialised ship types, including super-tankers and container ships, during later years of

* Tel.: C44 141 548 4465; fax: C44 141 552 2879. E-mail address: [email protected]

0029-8018/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2005.04.016

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the 20th century. Due to large size of these vessels, it has become increasingly difficult for many ports to accommodate primarily due to waterway restrictions. Also, the increases in the service speeds for new container ships and passenger liners, many areas of seas and channels, which were considered deep and safe for vessel navigation have been transferred into dangerous shallow water regions for operation of the new generation vessels. This has hence become an area of prime concern to the port authorities and operators, designers, and owners of the vessel alike. It is thus of paramount importance to understand the shallow water effects on the navigating vessel to ensure the safe limits of its speed and the resulting bottom clearance. This has led to the need for a more in-depth study of ship behaviour in waterways, restricted both in depth and width. Accurate prediction of this phenomenon for vessels operating in the restricted regions is essential for its safety from grounding. The ‘Go/No Go’ charts used by large tankers for the Strait of Mallaca, are results of such waterway restriction studies, which inform the master the safe speed limits at different regions and also how much the vessel sinks. The objective of the present study is to determine the under-keel clearance for an high speed vessel under different operating conditions, such as varying vessel speeds and waterway restrictions. This theoretical study will give information about safe vessel speed to prevent vessel grounding when it operates in restricted waterways.

2. Squat and its effects The explanation for squat, its effects and influencing parameters are well presented by Barrass (1979, 1997 and 1998). It is known that a two-dimensional body moving in a viscous fluid will not experience any vertical forces or trimming moments and will therefore not sink and trim further with forward motion. However, when the form becomes three-dimensional, pressure induced forces and pressure plus friction induced moments will result from forward speed and the body will react by sinking and trimming to maintain equilibrium. In the case of a ship moving in water, the body pushes water ahead as it moves forward and this water return down the sides and under the keel of the ship, which is termed as ‘back flow’. This flow causes pressure changes around the ship leading to a change in its draft. Also, due to the asymmetry of the pressure variation with respect to the principal transverse plane of the ship, it also trims. The combined effect of this is the reduced under-keel clearance of the vessel, and known as ‘ship squat’. While the vessel enters the shallow water region, the following changes are noticed to occur on the vessel presumably due to squat. 1. 2. 3. 4. 5.

increase in vessel’s wave-making reduction in speed of the vessel due to increase in resistance reduction in propeller rpm compared to that deep water condition reduction in course-changing ability of the vessel vibrations on the vessel due to the entrained water effect causing the natural hull frequency to become resonant with another frequency

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Sinkage and trim are both influenced by environmental conditions (shallow water, restricted channels, density layers, thermoclines, etc) and hull conditions (form, type of propulsion, hull fittings, appendages, etc). The main factor causing an increase in squat, and consequently in draft, is increase of speed. Additionally, interaction between two or more ships are known to increase the squat considerably, and the same has been proved that when two ships are abreast, the squat of each vessel could be doubled Barrass (1997).

3. Background study The squat phenomena and its various aspects have been evaluated by a number of authors. Kreitner (1934) was the first to calculate sinkage from the basic equations of fluid mechanics. Havelock (1939) considered an elliptical hull sailing through deep water and analytically estimated its sinkage. But the study of sinkage in shallow water is practically more important due to probable grounding situation. Constantine (1961) considered the case of a ship travelling in a narrow and shallow channel. At low speeds the flow accelerates past of the ship, producing a pressure decrease beneath the hull and a downward sinkage of the ship. Once the ship reaches a certain speed, which depends on the dimensions of ship and channel, no such steady flow existed. Constantine predicted that a bore travelling ahead of the vessel would be formed at this critical speed, resulting in a new type of steady speed. At a higher speed (ie. at supercritical speed) the flow will be a decelerated one past the vessel, causing the vessel to rise in the water. Tuck (1966) developed a slender-body theory that was valid for shallow water of infinite lateral extent. The sinkage and trim expressions gave good results, but these fail for the case of the ship speed approaching the speed of the waves in shallow water where the expressions become singular. Tuck (1967) modified the above theory to suit for ships operating in finite width channels, where the beam of the ship is considered as small compared to the channel width. Dand and Ferguson (1973) presented similar derivations and applied them to rectangular channel sections. Naghdi and Rubin (1984) studied the squat problem using a two-dimensional approach and compared the results with model test data. As theoretical expressions are not handy for regular use of mariners, many researchers have come up with empirical formulae for the estimation of sinkage and trim of ships, mainly based on model test results. By applying certain empirical correctors, derived from model tests, Dand and Ferguson (1973) have modified the calculation method to be allowed for squat prediction in shallow water of unrestricted waterway width. This semiempirical method has reduced the complex phenomena of squat to a straightforward calculation and also takes into account effects of viscosity, propeller action and hull wave system on the sinkage and trim. Experimental results of Hooft (1974) show good agreement with the sinkage formula of Tuck and Taylor (1970), based on vessel displacement and length. But Huuska (1976) and Millward (1989,1990,1992) have reported that this formula has a tendency to underestimate experimental results. Empirical expressions for squat, depending on vessel-speed and h/T values, have been put forward by Eryuzhu and Hausser (1978); Eryuzhu et al (1994); Barrass (1979).

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The empirical expressions discussed above predict the squat correctly in the case of ship types, channel types and speed ranges for which experimental data are available. However, they cannot be expected to be universally applicable for all types of ships, all channels and speeds.

4. Methodology The methodology considers steady flow around ship where, the flow is conceptually altered by referring it with respect to that of the ship. That is, the vessel is viewed as being at rest in the infinitely long channel with the water far ahead and astern of the ship flowing undisturbed at a uniform velocity V equal to that of the ship. The inherent assumption is that the ship sinkage is equal to the change in water level. Squat is then calculated by using Bernoulli’s equation combined with the equation of the continuity of flow. The formulations adopted here for the squat estimation are based on the work of Dand and Ferguson (1973), where a ship with sectional area of S(x) and local beam B(x) in a channel having rectangular cross-section area S0 and width w.and depth h are considered as shown in Fig. 1. By considering only the ‘back flow’, having perturbation velocity v(x), and neglecting the perturbation velocities in the y and z directions, it is further assumed that the stream of water in the vicinity of the vessel is uniform and flows past the vessel with a velocity of V1(x)ZVCv(x). Then, the following relationship describes the conservation of energy and continuity in terms of depths, velocities and areas. VS0 Z V1 ðxÞfS0 C ½wKBðxÞzðxÞKSðxÞg

(1)

where 2 (x) is the local water surface level change and v(x) is the perturbation velocity at a particular ship cross section due to ship presence, both of which vary along the length of the hull. At the free surface, the application of Bernoulli’s equation yields ½V 2 Z ½V12 ðxÞ C gzðxÞ

(2)

where ‘g’ is the acceleration due to gravity. Local Beam B(x) Free Surface z=ζ

z y

Undisturbed Water Level

h Channel Cross Section Area So

w Fig. 1. Problem definition sketch.

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By combining Eqs. (1) and (2) we get the combined relationship as a function of the location x, as follows 

 2
2  Fnh BðxÞ V1 ðxÞ 3 Fnh BðxÞ V1 ðxÞ C1 Z 0 (3) 1K 1K K 1KmðxÞ C w V w V 2 2 pffiffiffiffiffi where m(x)ZS(x)/S0 is the local blockage ratio and Fnh Z V= gh is the depth Froude number. Eq. (3) is solved for any x-coordinate for a fixed value of Fnh when the local blockage ratio is known. To simulate restricted water ways, the channel is considered to be of finite width (comparable to vessel’s length) but large enough for ship breadth B(x) to be assumed very small compared to w, i.e., B(x)/wy0. The above equation then reduces further giving an approximate solution for the perturbation velocity v(x) as vðxÞ Z

VmðxÞ 2 1KmðxÞKFnh

_ is obtained from Eq. (2) as Further, the non-dimensional surface elevation zðxÞ
 zðxÞ v 0 ðxÞ 2 0 ZKFnh z 0 ðxÞ Z v ðxÞ 1 C h 2 where h is the at-rest water depth and v 0 (x)Zv(x)/V The force and moment acting on the ship hull are then given by ð F Z rgh z 0 ðxÞBðxÞdx

(4)

(5)

L

ð M Z rgh z 0 ðxÞ½BðxÞxdx

(6)

L

The moments are taken about the LCF and B(x) represents the half-breadth of the at-rest floating water-plane. To maintain equilibrium, the vessel will sink and trim. By considering sinkage and trim as small, hydrostatic equations are applied. The equations for force and moment are written as F Z rgAwp Sm

(7)

M Z rgIL t

(8)

where Awp is the water-plane area, Sm is the mean sinkage, IL is the longitudinal moment of inertia of the water-plane area about centre of floatation and t is the trim angle in radians. From Eqs. (3)–(6), the non-dimensional mean sinkage and trim are obtained as    Ð 0  Sm !100 h T L Ðz ðxÞBðxÞdx CS Z Z 100 (9) LPP T LPP L BðxÞdx

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   Ð 0 h L z ðxÞBðxÞdx CT Z t !100 Z 100 ðTÞ Ð 2 T L BðxÞx dx

(10)

The resultant sinkage at the fore and aft perpendiculars of the vessel will be SFP Z

LPP ð2CS KCT Þ 200

(11a)

SAP Z

LPP ð2CS C CT Þ 200

(11b)

An ‘Effective Width’ concept put forward by Tuck (1967) to take care of the three for dimensional state of the squat problem resulted in the application of correctors, aðwÞ  for trim, which were derived from model tests and are given as sinkage and bðwÞ  S1 CSn Z aðwÞC

(12a)

 T1 CTn Z bðwÞC

(12b)

where the effective width, w is given as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w 2 w Z 1KFnh LPP

(13a)

The subscript ‘1’ in Eqs. (12a) and (12b) refers to values calculated by using Eqs. (9) and (10) in the fictitious channel cross-section and values with subscript ‘n’ are for naked towed hull in shallow water of infinite width. The correction factors are given as  Z aðwÞ

1 1 C dCS

(13b)

 Z bðwÞ

1 1 C dCT

(13c)

where  dCT Z 0:056K0:714w;  w Z 0:975LPP dCS Z 1:196K1:444w; Viscous effects on squat become considerable in shallow waters and the same are represented to some extent by these correction factors (Dand and Ferguson, 1973). The effect of self-propulsion on mean sinkage and trim is approximated by a further correction deducted from model data: CSP Z 1:1CSn

(14a)

  dC CTP Z 1 C Tn CTn CTn

(14b)

The trim correction factors due to propeller action are obtained from the graphs given by Dand (1973).

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Table 1 Main particulars of model bulk carrier Length between perpendiculars, LPP

Breadth at WL, B

Draft, T

Block coefficient, CB

3.048 m

0.518 m

0.174 m

0.80

(a) 0.0

Sinkage [cm]

–0.2 SAP SFP SAP (Exp) SFP (Exp) S UKC

–0.4 –0.6 –0.8 –1.0 –1.2 0

10

20

30

40

50

60

V [cm/s]

(b) 0.0

Sinkage [cm]

–0.4 –0.8 SAP SFP SAP (Exp) S FP (Exp) S UKC

–1.2 –1.6 –2.0 0

10

20

30

40

50

60

70

V [cm/s]

Sinkage [cm]

(c) 0.0 –1.0 SAP SFP SAP(Exp) SFP(Exp) SUKC

–2.0

–3.0

–4.0

0

10

20

30

40

50

60

70

80

90

V [cm/s] Fig. 2. (a) Sinkage at AP and FP of Bulk Carrier model for W/LZ1.5 and h/TZ1.05. (b) Sinkage at AP and FP of Bulk Carrier model for W/LZ1.5 and h/TZ1.10. (c) Sinkage at AP and FP of Bulk Carrier model for for W/LZ 1.5 and h/TZ1.20.

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Table 2 Main particulars of the high speed craft Length between perpendiculars, LPP

Breadth at WL, B

Draft, T

Block coefficient, CB

84.00 m

14.60 m

2.22 m

0.467

5. Numerical examples A computer program based on the above mathematical formulation has been developed to study the sinkage of ships under various operating conditions. To analyse the squat effect on various vessels two ships, a bulk carrier and a passenger ferry (High Speed monohull vessel) are considered. The passenger ferry has a finer form (CBZ0.467) compared to

Sinkage [m]

0.0 0.0 0.0 –0.1

SAP SFP S UKC

–0.1 –0.1 –0.1 0

1

2

3

4

5

V[knots] 0.00

Sinkage [m]

–0.02 –0.04 –0.06

SAP SFP SUKC

–0.08 –0.10 –0.12 0

1

2

3

4

5

V[knots] 0.00

Sinkage [m]

–0.02 –0.04 S A P

–0.06

S F P S U K C

–0.08 –0.10 –0.12 0

1

2

3

4

5

V[knots] Fig. 3. (a) Sinkage of High Speed Craft for h/TZ1.05 and W/LZ0.5. (b) Sinkage of High Speed Craft for h/TZ 1.05 and W/LZ1.0. (c) Sinkage of High Speed Craft for h/TZ1.05 and W/L O 3.0.

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that of the bulk carrier (CBZ0.81). The bulk carrier, for which results from model tests carried out in the Ship Hydrodynamics Lab of the University of Glasgow exist, has been used to validate the theoretical ones. Thereafter, studies on the squat effects of the HSC operating in Barcelona region have been carried out and presented in this paper. 5.1. Squat estimation for bulk carrier

(a)

0.00

Sinkage [m]

The model used by Ferguson (1978) for the tank tests at the Hydrodynamics Laboratory of the University of Glasgow is that of a Bulk Carrier and the model particulars are as in Table 1. The sinkage values measured from the model tests are compared with the theoretical ones in Fig. 2a–c.

–0.05 SAP SFP SUKC

–0.10 –0.15 –0.20 –0.25 0

1

2

3

4

5

6

4

5

6

4

5

6

V [knots]

Sinkage [m]

(b) 0.00 –0.04 –0.08 –0.12 –0.16 –0.20 –0.24

SAP SFP SUKC

0

1

2

3

V [knots]

(c)

0.00

Sinkage [m]

–0.04 –0.08 SAP SFP SUKC

–0.12 –0.16 –0.20 –0.24 0

1

2

3

V [knots] Fig. 4. (a) Sinkage of High Speed Craft for h/TZ1.10 and W/LZ0.5. (b) Sinkage of High Speed Craft for h/TZ 1.10 and W/LZ1.0. (c) Sinkage of High Speed Craft for h/TZ1.10 and W/L O 3.0.

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5.2. Squat estimation for high speed craft A mono-hull passenger ferry operating in Barcelona region is considered for the squat study. The main particulars of the ferry areas in Table 2. The estimated sinkages at the aft and fore perpendiculars of the vessel for different water-depth and channel-width conditions are shown in Figs. 3–8. Three dimensional surface representations of the same length are shown in Figs. 9–11.

Sinkage [m]

(a) 0.0 –0.1 SAP SFP SUKC

–0.2 –0.3 –0.4 –0.5 0

1

2

3

4

5

6

V [knots]

Sinkage [m]

(b) 0.1 0.0 –0.1 SAP SFP SUKC

–0.2 –0.3 –0.4 –0.5 0

1

2

3

4

5

6

7

V [knots]

(c) 0.2 Sinkage [m]

0.1 0.0 –0.1 SAP SFP S UKC

–0.2 –0.3 –0.4 –0.5 0

1

2

3

4

5

6

7

V [knots] Fig. 5. (a) Sinkage of High Speed Craft for h/TZ1.20 and W/LZ0.5. (b) Sinkage of High Speed Craft for h/TZ 1.20 and W/LZ1.0. (c) Sinkage of High Speed Craft for h/TZ1.20 and W/L O 3.0.

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Sinkage [m]

(a) 0.0 –0.1 –0.2 –0.3 –0.4 –0.5 –0.6 –0.7

SAP SFP SUKC

0

1

2

3

4

5

6

V [knots]

(b) 0.1 0.0

Sinkage [m]

–0.1 –0.2

SAP SFP SUKC

–0.3 –0.4 –0.5 –0.6 –0.7

0

1

2

3

4

5

6

7

V [knots] (c) 0.2 Sinkage [m]

0.0 –0.2

SAP SFP SUKC

–0.4 –0.6 –0.8

0

1

2

3

4

5

6

7

V [knots] Fig. 6. (a) Sinkage of High Speed Craft for h/TZ1.30 and W/LZ0.5. (b) Sinkage of High Speed Craft for h/TZ 1.30 and W/LZ1.0. (c) Sinkage of High Speed Craft for h/TZ1.30 and W/L O 3.0.

6. Results and discussion The numerical problems considered in the present study is discussed in the previous section. The results of each example are discussed separately below. Hydrodynamic model test results of a Bulk Carrier (Ferguson, 1978) are compared with the numerical values. The model test results are available only for a few cases of water-depth to draft ratios and a tank-width to vessel-length ratio of 1.5. This is due to the limitations of the available experimental facilities. Then the passenger ferry is considered for squat analysis, where the sub-critical speed range is used.

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Sinkage (m)

376 0.0 –0.1 –0.2 –0.3 –0.4 –0.5 –0.6 –0.7 –0.8 –0.9

SAP SFP S UKC

0

1

2

3

4

5

6

7

V (knots)

Sinkage (m)

0.2 0.0 –0.2

SAP SFP SUKC

–0.4 –0.6 –0.8 –1.0

0

1

2

3

4

5

6

7

8

5

6

7

8

V (knots) 0.2

Sinkage [m]

0.0 –0.2 SAP SFP SUKC

–0.4 –0.6 –0.8 –1.0 0

1

2

3

4

V (knots) Fig. 7. (a) Sinkage of HSVA Craft for h/TZ1.40 and W/LZ0.5. (b) Sinkage of HSVA Craft for h/TZ1.40 and W/LZ1.0. (c) Sinkage of HSVA Craft for h/TZ1.40 and W/LZ3.0.

6.1. Experimental validation of the numerical method The experimental test results of squat obtained from the model tests of a bulk carrier, carried out in the towing tank facility of the University of Glasgow are compared with the theoretical results. The results are shown in Fig. 2a–c. As the tank width is 4.57 m, the W/L (tank-width to model-length) ratio is 1.5. Experimental results pertain to the sinkages of the ship measured at both aft and fore perpendiculars (SAP, SFP) for water depth to model draft ratios (h/T) of 1.05, 1.1 and 1.2 for various model speeds in self-propulsion condition. The available static under-keel clearance (SUKC) in each condition is also shown in the plots. From the graphs (Fig. 2a–c) it can be seen that the sinkage at the fore perpendicular (SFP) is greater than that at the aft perpendicular (SAP) of the model at any particular

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Sinkage [m]

(a) 0.2 0.0 –0.2 –0.4 –0.6 –0.8 –1.0 –1.2

SAP SFP SUKC

0

1

2

3 4 V [knots]

5

6

7

Sinkage [m]

(b) 0.4 0.2 0.0 –0.2 –0.4 –0.6 –0.8 –1.0 –1.2

SAP SFP SUKC

0

1

2

3

4 V [knots]

5

6

7

8

Sinkage [m]

(c) 0.4 0.2 0.0 –0.2 –0.4 –0.6 –0.8 –1.0 –1.2

SAP SFP SUKC

0

1

2

3

4 5 V [knots]

6

7

8

9

Fig. 8. (a) Sinkage of High Speed Craft for h/TZ1.50 and W/LZ0.5. (b) Sinkage of High Speed Craft for h/TZ 1.50 and W/LZ1.0. (c) Sinkage of High Speed Craft for h/TZ1.50 and W/LO3.0.

model speed. This shows that the model trims by bow. The same trend is noticed for all the h/T values. From these figures it is also evident that the sinkage is more at lower h/T value compared to a higher h/T value for the same speed. That is, the sinkage increases as the under-keel clearance decreases. The model experimental measurements have shown a slightly lower sinkage compared to the numerical results for h/TZ1.05, where as for h/TZ1.1 and 1.2 the agreement between these two are generally good. This is probably due to neglect of viscosity effects which will be dominant in shallow water condition combined with lower speeds, whereas non linearity influences are more at higher speeds. Having validated the present numerical results with experimental results, the systematic parametric numerical investigation was carried out on the HSC model.

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(a)

Sinkage (m)

0.00 –0.10

1.50 1.45 1.40 1.35 1.30 1.25 h/T 1.20

–0.20 –0.30 –0.40

1.15 1.10

–0.50 0

2

3

1.05

4

5

V(knots)

–0.100-0.000 –0.200--0.100 –0.300--0.200 –0.400--0.300 –0.500--0.400

(b)

0.00

Sinkage (m)

–0.05

1.50 1.45 1.40 1.35 1.30 1.25 h/T 1.20

–0.10 –0.15 –0.20 –0.25

1.15

–0.30

0

2

3

V(knots)

4

5

1.05

1.10

–0.050-0.000 –0.100--0.050 –0.150--0.100 –0.200--0.150 –0.250--0.200 –0.300--0.250

Fig. 9. (a) 3D representation of the sinkage variations at AP of the High Speed Craft at W/LZ0.5. (b) 3D representation of the sinkage variations at FP of the High Speed Craft at W/LZ0.5.

6.2. High speed craft As an application problem the ferry operating in Barcelona, High Speed Craft, has been considered Varyani and Krishnankutty (2002). As the information such as the water depth and waterway restrictions and the vessel speeds in these regions are not readily available, values for h/T and W/L are selected in line with the previous examples to perform the squat calculations for the vessel. Figs. 3–8 show the graphical representation of the sinkage values over a range of vessel speed at the AP and FP of the vessel. As the vessel is finer in form with a CB of 0.467 and the longitudinal centre of buoyancy (LCB) is aft of mid-length of the vessel, it trims by the stern in all non-zero speeds. Three values for the waterway width are considered here for analysis, such as half of the vessel length, same as vessel length and much higher than vessel length (unrestricted width). Due

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(a) 0.0

Sinkage [m]

–0.2 –0.4

1.45 1.35

–0.6 1.25

–0.8

h/T

1.15 –1.0 0

2

3

4

–0.200-0.000 –0.400--0.200 –0.600--0.400 –0.800--0.600 –1.000--0.800

1.05

5

6

V [knots]

(b)

Sinkage [m]

0.00 –0.02 1.45

–0.04

1.35

–0.06

1.25

–0.08 –0.10

1.15 0

2

3

V [knots]

4

1.05 5

h/T

–0.020-0.000 –0.040--0.020 –0.060--0.040 –0.080--0.060 –0.100--0.080

Fig. 10. (a) 3D representation of the sinkage variations at AP of the High Speed Craft at W/LZ1.0. (b) 3D representation of the sinkage variations at FP of the High Speed Craft at W/LZ1.0.

to the blockage effect, the sinkage of the vessel in W/LZ0.5 case is considerably more than that in W/LZ1.0 case, at a particular vessel speed (see Figs. 3–8). But while comparing sinkage in W/LZ1.0 case with that in W/LO3 case, the values and pattern remain almost same. This means that the blockage effect is significant for the ferry only when W/L is less than 1.0. Almost similar performance has been observed for the depth conditions of h/TZ1.05, 1.10 and 1.20 considered here. Fig. 3A shows that the vessel grounds at a speed of 3.5 knots, when the width of the waterway is half the ferry length, where as the grounding occurs only at a speed of about 4.7 knots when the waterway width is equal to or greater than the vessel length. As expected grounding happens only at a higher speed (4.3 knots for W/LZ0.5, 5.7 knots for W/LO1.0) for deeper water (h/TZ1.1) condition, see Fig. 4A–C. Figs. 5–8 representing the sinkage effects at h/T values from 1.2 to 1.5, with an increment of 0.1, underline the above phenomenon. From Figs. 5 to 8, for W/LZ1.0 and W/LO3 and for a vessel speed greater than 6.0 knots, there appears to be a sudden increase in sinkage at the stern. This eventually leads to a higher emergence of the bow. The vessel has a fine form with CBZ0.467 much less than

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(a)

Sinkage [m]

0.0 –0.2 –0.4

1.45 1.35

–0.6 1.25

–0.8 –1.0 0

2

3

1.05

4

5

–0.200-0.000 –0.400--0.200 –0.600--0.400 –0.800--0.600 –1.000--0.800

6

V [knots]

Sinkage [m]

(b)

h/T

1.15

0.00 –0.02

1.45 1.35

–0.04 1.25

–0.06

h/T

1.15 –0.08 0

2

3

V [knots]

4

1.05 5

–0.020-0.000 –0.040--0.020 –0.060--0.040 –0.080--0.060

Fig. 11. (a) 3D representation of the sinkage variations at AP of the High Speed Craft at W/L O3. (b) 3D representation of the sinkage variations at FP of the High Speed Craft at W/L O3.

0.7 and LCB quite aft of the mid-length of the vessel. Hence a higher trim at the stern is expected in the sub-critical speed range of the vessel. This observation of sudden increase in the trim of the vessel, from about a speed of 6.0 knots obtained from the present software based on the theory explained in Section 2, need to be substantiated with model test results for the vessel. It is also of interest to notice that such behaviour is less pronounced in restricted waterway regions (W/LZ0.5) compared to wider waterways. Fig. 9a and b give a three-dimensional surface representation of the sinkage variations at AP and FP of the vessel at W/LZ0.5. Figs. 10 and 11 have similar values and patterns, which explicitly reveal the non-influence of waterway width when it is greater than the vessel’s length on the sinkage. 7. Conclusions The vessel considered here for squat studies are of different types, where the bulk carrier of full form (CBZ0.81) is considered for validation and the high speed vessel of

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fine form one (CBZ0.461) is considered for further studies. As expected the bulk carrier trims by the head and the passenger vessel trims by the stern. The following conclusions are drawn from the present studies: † Numerical results estimate reasonably accurate results for higher channel depth to ship draft (h/T) ratios but overestimate sinkage at lower h/T ratios, probably due to neglect of viscosity. † The effects of squat are magnified in shallow water and restricted waterways. Blockage effects on the vessel are found to be significant when W/L!1. † The speed at which vessel grounding occurs increases with W/L ratio. Increases in vessel’s speed have a much greater influence on the sinkage and trim of the vessel than the waterway restrictions thereby increasing the chances of vessel grounding. † Numerical results from the High Speed Craft indicate sinkage at the stern accompanied by a sudden emergence of bow for speeds greater than 6.0 knots and W/LR1. However, the same need to be substantiated further through model tests on the High Speed Craft.

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